# Properties

 Label 2205.2.a.i Level $2205$ Weight $2$ Character orbit 2205.a Self dual yes Analytic conductor $17.607$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$17.6070136457$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + q^{5} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + q^5 - 3 * q^8 $$q + q^{2} - q^{4} + q^{5} - 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{16} + 2 q^{17} - 4 q^{19} - q^{20} + 4 q^{22} + q^{25} + 2 q^{26} + 2 q^{29} + 5 q^{32} + 2 q^{34} - 10 q^{37} - 4 q^{38} - 3 q^{40} + 10 q^{41} + 4 q^{43} - 4 q^{44} + 8 q^{47} + q^{50} - 2 q^{52} + 10 q^{53} + 4 q^{55} + 2 q^{58} - 4 q^{59} + 2 q^{61} + 7 q^{64} + 2 q^{65} + 12 q^{67} - 2 q^{68} + 8 q^{71} - 10 q^{73} - 10 q^{74} + 4 q^{76} - q^{80} + 10 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} - 12 q^{88} - 6 q^{89} + 8 q^{94} - 4 q^{95} - 2 q^{97}+O(q^{100})$$ q + q^2 - q^4 + q^5 - 3 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^16 + 2 * q^17 - 4 * q^19 - q^20 + 4 * q^22 + q^25 + 2 * q^26 + 2 * q^29 + 5 * q^32 + 2 * q^34 - 10 * q^37 - 4 * q^38 - 3 * q^40 + 10 * q^41 + 4 * q^43 - 4 * q^44 + 8 * q^47 + q^50 - 2 * q^52 + 10 * q^53 + 4 * q^55 + 2 * q^58 - 4 * q^59 + 2 * q^61 + 7 * q^64 + 2 * q^65 + 12 * q^67 - 2 * q^68 + 8 * q^71 - 10 * q^73 - 10 * q^74 + 4 * q^76 - q^80 + 10 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 12 * q^88 - 6 * q^89 + 8 * q^94 - 4 * q^95 - 2 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 1.00000 0 0 −3.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.2.a.i 1
3.b odd 2 1 735.2.a.c 1
7.b odd 2 1 45.2.a.a 1
15.d odd 2 1 3675.2.a.j 1
21.c even 2 1 15.2.a.a 1
21.g even 6 2 735.2.i.e 2
21.h odd 6 2 735.2.i.d 2
28.d even 2 1 720.2.a.c 1
35.c odd 2 1 225.2.a.b 1
35.f even 4 2 225.2.b.b 2
56.e even 2 1 2880.2.a.bc 1
56.h odd 2 1 2880.2.a.y 1
63.l odd 6 2 405.2.e.c 2
63.o even 6 2 405.2.e.f 2
77.b even 2 1 5445.2.a.c 1
84.h odd 2 1 240.2.a.d 1
91.b odd 2 1 7605.2.a.g 1
105.g even 2 1 75.2.a.b 1
105.k odd 4 2 75.2.b.b 2
140.c even 2 1 3600.2.a.u 1
140.j odd 4 2 3600.2.f.e 2
168.e odd 2 1 960.2.a.a 1
168.i even 2 1 960.2.a.l 1
231.h odd 2 1 1815.2.a.d 1
273.g even 2 1 2535.2.a.j 1
336.v odd 4 2 3840.2.k.r 2
336.y even 4 2 3840.2.k.m 2
357.c even 2 1 4335.2.a.c 1
399.h odd 2 1 5415.2.a.j 1
420.o odd 2 1 1200.2.a.e 1
420.w even 4 2 1200.2.f.h 2
483.c odd 2 1 7935.2.a.d 1
840.b odd 2 1 4800.2.a.bz 1
840.u even 2 1 4800.2.a.t 1
840.bm even 4 2 4800.2.f.c 2
840.bp odd 4 2 4800.2.f.bf 2
1155.e odd 2 1 9075.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 21.c even 2 1
45.2.a.a 1 7.b odd 2 1
75.2.a.b 1 105.g even 2 1
75.2.b.b 2 105.k odd 4 2
225.2.a.b 1 35.c odd 2 1
225.2.b.b 2 35.f even 4 2
240.2.a.d 1 84.h odd 2 1
405.2.e.c 2 63.l odd 6 2
405.2.e.f 2 63.o even 6 2
720.2.a.c 1 28.d even 2 1
735.2.a.c 1 3.b odd 2 1
735.2.i.d 2 21.h odd 6 2
735.2.i.e 2 21.g even 6 2
960.2.a.a 1 168.e odd 2 1
960.2.a.l 1 168.i even 2 1
1200.2.a.e 1 420.o odd 2 1
1200.2.f.h 2 420.w even 4 2
1815.2.a.d 1 231.h odd 2 1
2205.2.a.i 1 1.a even 1 1 trivial
2535.2.a.j 1 273.g even 2 1
2880.2.a.y 1 56.h odd 2 1
2880.2.a.bc 1 56.e even 2 1
3600.2.a.u 1 140.c even 2 1
3600.2.f.e 2 140.j odd 4 2
3675.2.a.j 1 15.d odd 2 1
3840.2.k.m 2 336.y even 4 2
3840.2.k.r 2 336.v odd 4 2
4335.2.a.c 1 357.c even 2 1
4800.2.a.t 1 840.u even 2 1
4800.2.a.bz 1 840.b odd 2 1
4800.2.f.c 2 840.bm even 4 2
4800.2.f.bf 2 840.bp odd 4 2
5415.2.a.j 1 399.h odd 2 1
5445.2.a.c 1 77.b even 2 1
7605.2.a.g 1 91.b odd 2 1
7935.2.a.d 1 483.c odd 2 1
9075.2.a.g 1 1155.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{11} - 4$$ T11 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T - 2$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T + 10$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T - 2$$
$67$ $$T - 12$$
$71$ $$T - 8$$
$73$ $$T + 10$$
$79$ $$T$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 2$$